I just downsized my office and moved a large copier into my home. The copier has a NEMA 5-20P plug. I only have NEMA 5-15P standard receptacles. Do I need to run a dedicated circuit or can I change the recptacle and the 15 amp circuit in the main box without having to hire an electrician to install a new circuit?

A complete 20 amp circuit is needed, to include the NEMA 5-20 receptacle, #12 wire for the circuit wiring, and the 20 amp circuit breaker.

Is it possible that the receptacle already has #12 wire? How would I tell?

Tkrussel, there are also 5 other NEMA 5-15 receptacles on the sale circuit. If the wiring is #12, would it be OK to leave them on the 20 amp circuit?

If you can be totally sure the WHOLE circuit is wired with #12 then yes, the breaker can be changed to 20A. This is the only situation where this is true.
Are you also sure you will not need to use those other receptacles while using the copier? Big copiers like that are hogs and will near max out a 20A circuit while in use.